Notice tat this result makes no assumptions about the delay \(L\)  The delay could be large enough \((L\geqslant T)\)  to completely inhibit adaptivity during experimentation or short enough (\(L=1\) ) that there are no explicit delays in reward observations. The term \(\iota \)  in the bound above comes from applying a matrix concentration inequality to control the impact of the algorithm's action randomization. The bound applied there is crude and that term can almost certainly be tightened. The conditional entropy \(\mathbb {H}( I^* \mid H_{T}^+ )\)  is always smaller than the entropy term \(\mathbb {H}( I^*)\)  that appears in [1]}. The conditional entropy is upper bounded by \(\log (k)\) , but it can be much smaller if the decision-maker is likely to be confident about the identity of the best arm at the end of the problem. The requirement that \(\beta _t \geqslant 1/2\)  is satisfied by the default recommendation of using an unbiased coin (\(\beta _t=1/2\) ). If this is relaxed, the result would instead depend on \((\min _{t} \beta _t)^{-1}\) .
